Learning Objectives for Section 4'2

1
Learning Objectives for Section 4.2

• Systems of Linear Equations and Augmented
  Matrices
• The student will be able to use terms associated
  with matrices.
• The student will be able to set up and solve the
  augmented matrix associated with a linear system
  in two variables.
• The student will be able to identify the three
  possible matrix solution types for a linear
  system in two variables.

2
Matrix Methods

• It is impractical to solve more complicated
  linear systems by hand. Computers and calculators
  now have built in routines to solve larger and
  more complex systems. Matrices, in conjunction
  with graphing utilities and or computers are used
  for solving more complex systems. In this
  section, we will develop certain matrix methods
  for solving two by two systems.

3
Matrices
Since this matrix has 3 rows and 3 columns, the
dimensions of the matrix are 3 x 3.

• A matrix is a rectangular array of numbers
  written within brackets. Here is an example of a
  matrix which has three rows and three columns
  The subscripts give the address of each entry
  of the matrix. For example the entry a23 is found
  in the second row and third column

Each number in the matrix is called an element.
4
Matrix Solution of Linear Systems

• When solving systems of linear equations, we can
  represent a linear system of equations by an
  augmented matrix, a matrix which stores the
  coefficients and constants of the linear system
  and then manipulate the augmented matrix to
  obtain the solution of the system.

• Example
• x 3y 5
• 2x y 3

The augmented matrix associated with the above
system is
5
Generalization

• Linear system

• Associated augmented matrix

6
Operations that Produce Row-Equivalent Matrices

• 1. Two rows are interchanged
• 2. A row is multiplied by a nonzero constant
• 3. A constant multiple of one row is added to
  another row

7
Augmented Matrix MethodExample 1

• Solve
• x 3y 5
• 2x y 3
• 1. Augmented system
• 2. Eliminate 2 in 2nd row by row operation
• 3. Divide row two by -7 to obtain a coefficient
  of 1.
• 4. Eliminate the 3 in first row, second position.
  
• 5. Read solution from matrix

R2
8
Augmented Matrix MethodExample 2

• Solve
• x 2y 4
• x (1/2)y 4
• Eliminate fraction in second equationby
  multiplying by 2
• Write system as augmented matrix.
• Multiply row 1 by -2 and add to row 2
• Divide row 2 by -3
• Multiply row 2 by -2 and add to row 1.
• Read solution x 4, y 0
• (4,0)

9
Augmented Matrix MethodExample 3

• Solve
• 10x - 2y 6
• -5x y -3
• 1. Represent as augmented matrix.
• 2. Divide row 1 by 2
• 3. Add row 1 to row 2 and replace row 2 by sum
• 4. Since 0 0 is always true, we have a
  dependent system. The two equations are
  identical, and there are infinitely many
  solutions.

10
Augmented Matrix MethodExample 4

• Solve
• Rewrite second equation
• Add first row to second row
• The last row is the equivalent of 0x 0y -5
• Since we have an impossible equation, there is no
  solution. The two lines are parallel and do not
  intersect.

11
Possible Final Matrix Forms for a Linear System
in Two Variables
Form 1 Unique Solution
(Consistent and Independent)
Form 2 Infinitely Many Solutions
(Consistent and Dependent)
Form 3 No Solution (Inconsistent)